262 MATHEMATICS. 6. 3. 1. 1. , 1, , 0,. 2. 2. 2. −. −. − .... 1. (b), (d). In all these
cases data can be divided into class intervals. 2. Shopper. Tally marks. Number.
W.
ANSWERS
261
ANSWERS
EXERCISE 1.1 1. (i) 2 2. (i) 4. (i) 5. (i) (iii) 6.
−2 8
(ii)
−11 28
(ii)
5 9
(iii)
−6 5
−1 −19 (ii) (iii) 5 13 13 1 is the multiplicative identity Multiplicative inverse
− 96 91
2 9
(v)
19 6
56 5 (v) 15 2 (ii) Commutativity
(iv)
7. Associativity
9. Yes, because 0.3 × 3
(vi) –1
8. No, because the product is not 1.
1 3 10 × =1 = 3 10 3
10. (i) 0
(ii) 1 and (–1)
(iii) 0
11. (i) No
(ii) 1, –1
(iii)
(vi)
(iv)
−1 5
(iv) x
(v) Rational number
positive
EXERCISE 1.2 1. (i)
(ii)
2.
3. Some of these are 1, 4.
−1 1 , 0, –1, 2 2
−7 − 6 −5 − 4 −3 −2 −1 1 2 , , , , , , , 0, ... , , (There can be many more such rational numbers) 20 20 20 20 20 20 20 20 20
5. (i)
41 42 43 44 45 −8 −7 1 2 , , , , , , 0, , (ii) 60 60 60 60 60 6 6 6 6 (There can be many more such rational numbers)
(iii)
9 10 11 12 13 , , , , 32 32 32 32 32
262
MATHEMATICS
3 −1 1 6. − , − 1, , 0, (There can be many more such rational numbers) 2 2 2
7.
97 98 99 100 101 102 103 104 105 106 , , , , , , , , , 160 160 160 160 160 160 160 160 160 160 (There can be many more such rational numbers)
EXERCISE 2.1 1. x = 9
2. y = 7
3. z = 4
7. x = 27
8. y = 2.4
9. x =
25 7
4. x = 2
5. x = 2
3 2
11. p = –
10. y =
6. t = 50 4 3
12. x = –
8 5
EXERCISE 2.2 3 2 2. length = 52 m, breadth = 25 m 3. 1 cm 4 5 5. 45, 27 6. 16, 17, 18 7. 288, 296 and 304 9. Rahul’s age: 20 years; Haroon’s age: 28 years 10. 48 students 11. Baichung’s age: 17 years; Baichung’s father’s age: 46 years;
1.
12. 5 years
Baichung’s grandfather’s age = 72 years
4. 40 and 55 8. 7, 8, 9
13 −
1 2
14. Rs 100 → 2000 notes; Rs 50 → 3000 notes; Rs 10 → 5000 notes 15. Number of Re 1 coins = 80; Number of Rs 2 coins = 60; Number of Rs 5 coins = 20 16. 19
EXERCISE 2.3 1. x = 18
2. t = –1
3. x = –2
7. x = 40
8. x = 10
9. y =
7 3
4. z =
3 2
10. m =
5. x = 5
4 5
EXERCISE 2.4 1. 4
2. 7, 35
3. 36
4. 26 (or 62)
5. Shobo’s age: 5 years; Shobo’s mother’s age: 30 years 6. Length = 275 m; breadth = 100 m
7. 200 m
9. Grand daughter’s age: 6 years; Grandfather’s age: 60 years 10. Aman’s age: 60 years; Aman’s son’s age: 20 years
8. 72
6. x = 0
ANSWERS
263
EXERCISE 2.5 27 10 7 6. m = 5
1. x =
2. n = 36
3. x = –5
7. t = – 2
8. y =
2 3
4. x = 8
5. t = 2
9. z = 2
10. f = 0.6
EXERCISE 2.6 1. x =
3 2
2. x =
35 33
3. z = 12
6. Hari’s age = 20 years; Harry’s age = 28 years
4. y = – 8 7.
5. y = –
4 5
13 21
EXERCISE 3.1 1. (a) 1, 2, 5, 6, 7 (b) 1, 2, 5, 6, 7 (c) 1, 2, 4 (d) 2 (e) 1, 4 2. (a) 2 (b) 9 (c) 0 3. 360°; yes. 4. (a) 900° (b) 1080° (c) 1440° (d) (n – 2)180° 5. A polygon with equal sides and equal angles. (i) Equilateral triangle (ii) Square (iii) Regular hexagon 6. (a) 60° (b) 140° (c) 140° (d) 108° 7. (a) x + y + z = 360° (b) x + y + z + w = 360°
EXERCISE 3.2 1. (a) 360° – 250° = 110° 2. (i)
360° = 40° 9
(b) 360° – 310° = 50° (ii)
360° = 24° 15
360 = 15 (sides) 4. Number of sides = 24 24 5. (i) No; (Since 22 is not a divisor of 360) (ii) No; (because each exterior angle is 180° – 22° = 158°, which is not a divisor of 360°). 6. (a) The equilateral triangle being a regular polygon of 3 sides has the least measure of an interior angle = 60°. (b) By (a), we can see that the greatest exterior angle is 120°.
3.
EXERCISE 3.3 1. (i) BC(Opposite sides are equal)
(ii) ∠ DAB (Opposite angles are equal)
264
MATHEMATICS
(iii) OA (Diagonals bisect each other) 2.
3.
4. 7. 8.
(iv) 180° (Interior opposite angles, since AB || DC ) (i) x = 80°; y = 100°; z = 80° (ii) x = 130°; y = 130°; z = 130° (iii) x = 90°; y = 60°; z = 60° (iv) x = 100°; y = 80°; z = 80° (v) y = 112°; x = 28°; z = 28° (i) Can be, but need not be. (ii) No; (in a parallelogram, opposite sides are equal; but here, AD ≠ BC). (iii) No; (in a parallelogram, opposite angles are equal; but here, ∠A ≠ ∠C). A kite, for example 5. 108°; 72°; 6. Each is a right angle. x = 110°; y = 40°; z = 30° (i) x = 6; y = 9 (ii) x = 3; y = 13; 9. x = 50°
10. NM || KL (sum of interior opposite angles is 180°). So, KLMN is a trapezium. 11. 60° 12. ∠P = 50°; ∠S = 90°
EXERCISE 3.4 1. (b), (c), (f), (g), (h) are true; others are false. 2. (a) Rhombus; square. (b) Square; rectangle 3. (i) A square is 4 – sided; so it is a quadrilateral. (ii) A square has its opposite sides parallel; so it is a parallelogram. (iii) A square is a parallelogram with all the 4 sides equal; so it is a rhombus. (iv) A square is a parallelogram with each angle a right angle; so it is a rectangle. 4. (i) Parallelogram; rhombus; square; rectangle. (ii) Rhombus; square (iii) Square; rectangle 5. Both of its diagonals lie in its interior. 6. AD || BC; AB || DC . So, in parallelogram ABCD, the mid-point of diagonal AC is O.
EXERCISE 5.1 1. (b), (d). In all these cases data can be divided into class intervals. 2.
Shopper
Tally marks
Number
W
|||| |||| |||| |||| |||| |||
28
M
|||| |||| ||||
15
B
||||
G
|||| |||| ||
5 12
ANSWERS
3.
Interval
Tally marks
Frequency
800 - 810
|||
3
810 - 820
||
2
820 - 830
|
1
830 - 840
|||| ||||
9
840 - 850
||||
5
850 - 860
|
1
860 - 870
|||
3
870 - 880
|
1
880 - 890
|
1
890 - 900
||||
4
Total
4. (i) 830 - 840 (iii) 20
(ii) 10
5. (i) 4 - 5 hours (iii) 14
(ii) 34
265
30
EXERCISE 5.2 1. (i) 200
(ii) Light music (iii) Classical - 100, Semi classical - 200, Light - 400, Folk - 300
2. (i) Winter (ii) Winter - 150°, Rainy - 120°, Summer - 90° 3.
(iii)
266
MATHEMATICS
4. (i) Hindi (ii) 30 marks
(iii) Yes
5.
EXERCISE 5.3 1. (a) Outcomes → A, B, C, D (b) HT, HH, TH, TT (Here HT means Head on first coin and Tail on the second coin and so on). 2. Outcomes of an event of getting (i) (a) 2, 3, 5 (b) 1, 4, 6 (ii) (a) 6 (b) 1, 2, 3, 4, 5 3. (a) 4. (i)
1 5 1 10
1 13 1 (ii) 2
(b)
4 7 2 (iii) 5
(c)
(iv)
9 10
3 4 ; probability of getting a non-blue sector = 5 5 1 1 6. Probability of getting a prime number = ; probability of getting a number which is not prime = 2 2 1 Probability of getting a number greater than 5 = 6 5 Probability of getting a number not greater than 5 = 6
5. Probability of getting a green sector =
EXERCISE 6.1 1. (i) 1 (ii) 4 (iii) 1 (iv) 9 (v) 6 (vii) 4 (viii) 0 (ix) 6 (x) 5 2. These numbers end with (i) 7 (ii) 3 (iii) 8 (iv) 2 (v) 0 (vii) 0 (viii) 0 3. (i), (iii) 4. 10000200001, 100000020000001 5. 1020304030201, 1010101012 6. 20, 6, 42, 43 7. (i) 25 (ii) 100 (iii) 144 8. (i) 1 + 3 + 5 + 7 + 9 + 11 + 13 (ii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 9. (i) 24 (ii) 50 (iii) 198
(vi) 9
(vi) 2
ANSWERS
267
EXERCISE 6.2 1. (i) 1024 (ii) 1225
(iii) 7396
(iv) 8649
2. (i) 6,8,10 (ii) 14,48,50
(iii) 16,63,65
(iv) 18,80,82
(v) 5041
(vi) 2116
(v) 88
(vi) 98
EXERCISE 6.3 1. (i) 1, 9 2. (i), (ii), (iii) 4. (i) 27 (vii) 77 5. (i) 7; 42 6. (i) 7; 6 7. 49
(ii) 4, 6 (ii) (viii) (ii) (ii)
(iii) 1, 9
(iv) 5
3. 10, 13 20 (iii) 42 (iv) 64 96 (ix) 23 (x) 90 5; 30 (iii) 7, 84 (iv) 3; 78 13; 15 (iii) 11; 6 (vi) 5; 23 8. 45 rows; 45 plants in each row
(v) 2; 54 (v) 7; 20 9. 900
(vi) 3; 48 (vi) 5; 18 10. 3600
(v) (xi) (v) (v) (v) (v)
(vi) 37 (xii) 30
EXERCISE 6.4 1. (i) 48 (ii) 67 (iii) 59 (vii) 76 (viii) 89 (ix) 24 2. (i) 1 (ii) 2 (iii) 2 3. (i) 1.6 (ii) 2.7 (iii) 7.2 4. (i) 2; 20 (ii) 53; 44 (iii) 1; 57 5. (i) 4; 23 (ii) 14; 42 (iii) 4; 16 6. 21 m 7. (a) 10 cm 8. 24 plants 9. 16 children
(iv) (x) (iv) (iv) (iv) (iv)
23 32 3 6.5 41; 28 24; 43 (b) 12 cm
57 56 3 5.6 31; 63 149; 81
EXERCISE 7.1 1. 2. 3. 4.
(ii) and (iv) (i) 3 (ii) 2 (i) 3 (ii) 2 20 cuboids
(iii) 3 (iii) 5
(iv) 5 (iv) 3
(v) 10 (v) 11
(iv) 30
(v) 25
(vi) 24
(iv) False
(v) False
(vi) False
EXERCISE 7.2 1. (i) 4 (ii) 8 (vii) 48 (viii) 36 2. (i) False (ii) True (vii) True 3. 11, 17, 23, 32
(iii) 22 (ix) 56 (iii) False
268
MATHEMATICS
EXERCISE 8.1 1. (a) 1 : 2
(b) 1 : 2000 (c) 1 : 10 2 2. (a) 75% (b) 66 % 3. 28% students 4. 25 matches 5. Rs 2400 3 6. 10%, cricket → 30 lakh; football → 15 lakh; other games → 5 lakh
EXERCISE 8.2 1. Rs 1,40,000 5. Gain of 2% 8. Rs 14,560
2. 80% 6. Rs 2,835 9. Rs 2,000
3. Rs 34.80 4. Rs 18,342.50 7. Loss of Rs 1,269.84 10. Rs 5,000
EXERCISE 8.3 1. (a) Amount = Rs 15,377.34; Compound interest = Rs 4,577.34 (b) Amount = Rs 22,869; Interest = Rs 4869 (c) Amount = Rs 70,304, Interest = Rs 7,804 (d) Amount = Rs 8,736.20, Interest = Rs 736.20 (e) Amount = Rs 10,816, Interest = Rs 816 2. Rs 36,659.70 3. Fabina pays Rs 362.50 more 4. Rs 43.20 5. (ii) Rs 63,600 (ii) Rs 67,416 6. (ii) Rs 92,400 (ii) Rs 92,610 7. (i) Rs 8,820 (ii) Rs 441 8. Amount = Rs 11,576.25, Interest = Rs 1,576.25 Yes. 9. Rs 4,913 10. (i) About 48,980 (ii) 59,535 11. 5,31,616 (approx) 12. Rs 38,640
EXERCISE 9.1 1.
Term
Coefficient
(i)
5xyz2 –3zy
5 –3
(ii)
1 x x2
1 1 1
(iii)
4x2y2 – 4x2y2z2 z2
4 –4 1
(iv)
(v)
(vi)
3 – pq qr – rp
3 –1 1 –1
x 2 y 2 –xy
1 2 1 2 –1
0.3a – 0.6ab 0.5b
0.3 – 0.6 0.5
ANSWERS
2. Monomials: 1000, pqr Binomials: x + y, 2y – 3y2, 4z – 15z2, p2q + pq2, 2p + 2q Trinomials :7 + y + 5x, 2y – 3y2 + 4y3, 5x – 4y + 3xy Polynomials that do not fit in these categories: x + x2 + x3 + x4, ab + bc + cd + da 3. (i) 0 (ii) ab + bc + ac (iii) – p2q2 + 4pq + 9 (iv) 2(l2 + m2 + n2 + lm + mn + nl) 4. (a) 8a – 2ab + 2b – 15 (b) 2xy – 7yz + 5zx + 10xyz 2 2 (c) p q – 7pq + 8pq – 18q + 5p + 28
EXERCISE 9.2 1. (i) 28p (ii) – 28p2 (iii) – 28p2q 2. pq; 50 mn; 100 x2y2; 12x3; 12mn2p 3.
(iv) –12p4
(v) 0
First monomial → Second monomial ↓
2x
–5y
3x 2
– 4xy
7x 2y
–9x2y2
2x
4x 2
–10xy
6x 3
–8x2y
14x3y
–18x3y2
–5y
–10xy
25y2
–15x2y
20xy2
3x 2
6x 3
–15x2y
9x 4
–12x3y
– 4xy
–8x 2y
20xy 2
–12x3y
16x2y2
–28x3y2 36x3y3
7x 2y
14x 3y
–35x2y2
21x 4y
–28x3y
49x4y2 – 63x4y3
–9x2y2
–18x3y2
45x2y3
4. (i) 105a7 (ii) 64pqr 5. (i) x2y2z 2 (ii) – a6
–27x4y2 36x3y2
(iii) 4x4y 4 (iii) 1024y6
–35x2y2 45x2y3 21x4y
–27x4y2
– 63x4y3 81x4y4
(iv) 6abc (iv) 36a2b2c2
(v) – m3n2p
EXERCISE 9.3 1. (i) (iv) 2. (i) (iii) (v)
4pq + 4pr 4a3 – 36a ab + ac + ad 6p3 – 7p2 + 5p a2bc + ab2c + abc2 3 3. (i) 8a50 (ii) − x 3 y3 5
a2b – ab2 (iii) 7a3b2 + 7a2b3 0 5x2y + 5xy2 – 25xy 4p4q2 – 4p2q4
(iii) – 4p4q4
(iv) x 10
−3 2 a3 + a2 + a + 5; (i) 5 (ii) 8 (iii) 4 2 2 2 2 2 p + q + r – pq – qr – pr (b) – 2x – 2y – 4xy + 2yz + 2zx 5l2 + 25ln (d) – 3a2 – 2b2 + 4c2 – ab + 6bc – 7ac
4. (a) 12x2 – 15x + 3; (b) 5. (a) (c)
(ii) (v) (ii) (iv)
(i) 66
(ii)
269
270
MATHEMATICS
EXERCISE 9.4 1. (i) (iv) 2. (i) (iv) 3. (i) (iv) (vii)
8x2 + 14x – 15 ax + 5a + 3bx + 15b 15 – x – 2x2 2p3 + p2q – 2pq2 – q3 x3 + 5x2 – 5x 4ac 2.25x2 – 16y2
(ii) 3y2 – 28y + 32 (v) 6p2q2 + 5pq3 – 6q4 (ii) 7x2 + 48xy – 7y2
(iii) 6.25l2 – 0.25m2 (vi) 3a4 + 10a2b2 – 8b4 (iii) a3 + a2b2 + ab + b3
(ii) a2b3 + 3a2 + 5b3 + 20 (v) 3x2 + 4xy – y2 (viii) a2 + b2 – c2 + 2ab
(iii) t3 – st + s2t2 – s3 (vi) x3 + y3
EXERCISE 9.5 1. (i) x2 + 6x + 9 (iv) 9a2 – 3a +
1 4
2
(vii) 36x – 49 (x) 2. (i) (iv) (vii) 3. (i) (iv) 4. (i) (iv) 6. (i) (v) (ix) 7. (i) 8. (i)
49a2 – 126ab + 81b2 x2 + 10x + 21 16x2 + 16x – 5 x2 y2 z2 – 6xyz + 8 b2 – 14b + 49 4 2 9 m + 2mn + n2 9 4 a4 – 2a2b2 + b4 41m2 + 80mn + 41n2 5041 27.04 9.975 200 10712
(ii) 4y2 + 20y + 25
(iii) 4a2 – 28a + 49
(v) 1.21m2 – 0.16
(vi) b4 – a4
2
2
(viii) a – 2ac + c
(ix)
x 2 3xy 9 y 2 + + 4 4 16
(ii) 16x2 + 24x + 5 (v) 4x2 + 16xy + 15y2
(iii) 16x2 – 24x + 5 (vi) 4a4 + 28a2 + 45
(ii) x2 y2 + 6xyz + 9z2
(iii) 36x4 – 60x2y + 25y2
(v) 0.16p2 + 0.04pq + 0.25q2
(vi) 4x2 y2 + 20xy2 + 25y2
(ii) (v) (ii) (vi)
40x 4p2 – 4q2 9801 89991
(ii) 0.08 (ii) 26.52
(iii) (vi) (iii) (vii)
98m2 + 128n2 a2b2 + b2c2 (vii) m4 + n4m2 10404 (iv) 996004 6396 (viii) 79.21
(iii) 1800 (iii) 10094
(iv) 84 (iv) 95.06
EXERCISE 10.1 1. (a) → (iii) → (iv) (b) → (i) → (v) (d) → (v) → (iii) (e) → (ii) → (i) 2. (a) (i) → Front, (ii) → Side, (iii) → Top (c) (i) → Front, (ii) → Side, (iii) → Top 3. (a) (i) → Top, (ii) → Front, (iii) → Side (c) (i) → Top, (ii) → Side, (iii) → Front (e) (i) → Front, (ii) → Top, (iii) → Side
(c) → (iv) → (ii) (b) (d) (b) (d)
(i) → Side, (ii) → Front, (iii) → Top (i) → Front, (ii) → Side, (iii) → Top (i) → Side, (ii) → Front, (iii) → Top (i) → Side, (ii) → Front, (iii) → Top
ANSWERS
271
EXERCISE 10.3 1. (i) No (ii) Yes (iii) Yes 2. Possible, only if the number of faces are greater than or equal to 4 3. only (ii) and (iv) 4. (i) A prism becomes a cylinder as the number of sides of its base becomes larger and larger. (ii) A pyramid becomes a cone as the number of sides of its base becomes larger and larger. 5. No. It can be a cuboid also 7. Faces → 8, Vertices → 6, Edges → 30 8. No
EXERCISE 11.1 1. (a) 4. 45000 tiles
2. Rs 17,875 5. (b)
3. Area = 129.5 m2; Perimeter = 48 m
EXERCISE 11.2 1. 0.88 m2 5. 45 cm2
3. 660 m2 4. 252 m2 7. Rs 810 8. 140 m 1 15 × (30 + 15) m 2 = 337.5 m 2 , 9. 119 m2 10. Area using Jyoti’s way = 2 × × 2 2 1 2 Area using Kavita’s way = × 15 × 15 + 15 × 15 = 337.5 m 2 11. 80 cm2, 96 cm2, 80 cm2, 96 cm2 2. 7 cm 6. 24 cm2, 6 cm
EXERCISE 11.3 1. (a) 2. 144 m 3. 10 cm 4. 11 m2 5. 5 cans 6. Similarity → Both have same heights. Difference → one is a cylinder, the other is a cube. The cube has larger lateral surface area 7. 440 m2 8. 322 cm 9. 1980 m2 10. 704 cm2
EXERCISE 11.4 1. 2. 3. 7.
(a) Volume (b) Surface area (c) Volume Volume of cylinder B is greater; Surface area of cylinder B is greater. 5 cm 4. 450 5. 1 m 6. 49500 L (i) 4 times (ii) 8 times 8. 30 hours
EXERCISE 12.1 1. (i)
1 9
(ii)
1 16
(iii) 32
272
2. (i)
MATHEMATICS
1 (– 4)3
3. (i) 5 4. (i) 250 7. (i)
(ii) 1 2 1 (ii) 60
(ii)
1 26
(iii) 29
(iii) (5)4
(iv)
(iv) 1
(v)
5. m = 2
1 (3)2
(v)
1 ( −14) 3
81 16 512 (ii) 125
6. (i) –1
625t 4 (ii) 55 2
EXERCISE 12.2 1. (i) 8.5 × 10– 12 (iv) 8.37 × 10– 9 2. (i) 0.00000302 (iv) 1000100000 3. (i) 1 × 10– 6 (iv) 1.275 × 10–5 4. 1.0008 × 102
(ii) (v) (ii) (v) (ii) (v)
9.42 × 10– 12 3.186 × 1010 45000 5800000000000 1.6 × 10–19 7 × 10–2
(iii) 6.02 × 1015 (iii) 0.00000003 (vi) 3614920 (iii) 5 × 10– 7
EXERCISE 13.1 1. No
2.
Parts of red pigment Parts of base
1 8
4 32
7 56
3. 24 parts 4. 700 bottles 5. 10– 4 cm; 2 cm 7. (i) 2.25 × 107 crystals (ii) 5.4 × 106 crystals 9. (i) 6 m (ii) 8 m 75 cm 10. 168 km
12 96
20 160
6. 21 m 8. 4 cm
EXERCISE 13.2 2. 4 → 25,000; 5 → 20,000; 8 → 12,500; 10 → 10,000; 20 → 5,000 Amount given to a winner is inversely proportional to the number of winners. 3. 8 → 45°, 10 → 36°, 12 → 30° (i) Yes (ii) 24° (iii) 9 4. 6 5. 4 6. 3 days 7. 15 boxes 1 10. (i) 6 days (ii) 6 persons 11. 40 minutes 8. 49 machines 9. 1 hours 2
1. (i), (iv), (v)
EXERCISE 14.1 1. (i) 12 (vii) 10
(ii) 2y (viii) x 2y 2
(iii) 14pq
(iv) 1
(v) 6ab
(vi) 4x
ANSWERS
2. (i) (v) (viii) 3. (i) (iv)
7(x – 6) 10 lm(2l + 3a) 4a(– a + b – c) (x + 8) (x + y) (5p + 3) (3q + 5)
(ii) (vi) (ix) (ii) (v)
6(p – 2q) (iii) 7a(a + 2) (iv) 4z(– 4 + 5z2) 5xy(x – 3y) (vii) 5(2a2 – 3b2 + 4c2) xyz(x + y + z) (x) xy(ax + by + cz) (3x + 1) (5y – 2) (iii) (a + b) (x – y) (z – 7) (1 – xy)
EXERCISE 14.2 1. (i) (v) 2. (i) (iv) (vii) 3. (i) (iv) (vii) 4. (i) (iii) (v) 5. (i)
(a + 4)2 (ii) (p – 5)2 (iii) (5m + 3)2 (iv) (7y + 6z)2 4(x – 1)2 (vi) (11b – 4c)2 (vii) (l – m)2 (viii) (a2 + b2)2 (2p – 3q) (2p + 3q) (ii) 7(3a – 4b) (3a + 4b) (iii) (7x – 6) (7x + 6) 3 16x (x – 3) (x + 3) (v) 4lm (vi) (3xy – 4) (3xy + 4) (x – y – z) (x – y + z) (viii) (5a – 2b + 7c) (5a + 2b – 7c) x(ax + b) (ii) 7(p2 + 3q2) (iii) 2x(x2 + y2 + z2) (m2 + n2) (a + b) (v) (l + 1) (m + 1) (vi) (y + 9) (y + z) (5y + 2z) (y – 4) (viii) (2a + 1) (5b + 2) (ix) (3x – 2) (2y – 3) (a – b) (a + b) (a2 + b2) (ii) (p – 3) (p + 3) (p2 + 9) (x – y – z) (x + y + z) [x2 + (y + z)2] (iv) z(2x – z) (2x2 – 2xz + z2) (a – b)2 (a + b)2 (p + 2) (p + 4) (ii) (q – 3) (q – 7) (iii) (p + 8) (p – 2)
EXERCISE 14.3 1. (i) 2. (i) (iv) 3. (i)
x3 (ii) – 4y 2 1 (5 x − 6) 3 1 2 ( x + 2 x + 3) 2 2x – 5 (ii) 5
4. (i) 5(3x + 5)
(iii) 6pqr
(iv)
2 2 x y 3
(v) –2a2b4
(ii) 3y4 – 4y2 + 5
(iii) 2(x + y + z)
(v) q3 – p3 (iii) 6y (ii) 2y(x + 5)
(iv) xy (v) 10abc 1 r ( p + q ) (iv) 4(y2 + 5y + 3) (iii) 2
(v) (x + 2) (x + 3) 5. (i) y + 2 (vi)
(ii) m – 16
3(3x – 4y)
(iii) 5(p – 4)
(iv) 2z(z – 2)
(v)
5 q( p − q) 2
(vii) 3y(5y – 7)
EXERCISE 14.4 1. 4(x – 5) = 4x – 20 4. x + 2x + 3x = 6x
2. x(3x + 2) = 3x2 + 2x 5. 5y + 2y + y – 7y = y
3. 2x + 3y = 2x + 3y 6. 3x + 2x = 5x
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7. (2x)2 + 4(2x) + 7 = 4x2 + 8x + 7 8. (2x)2 + 5x = 4x2 + 5x 9. (3x + 2)2 = 9x2 + 12x + 4 10. (a) (–3)2 + 5(–3) + 4 = 9 – 15 + 4 = –2 (b) (–3)2 – 5(–3) + 4 = 9 + 15 + 4 = 28 (c) (–3)2 + 5(–3) = 9 – 15 = – 6 11. (y – 3)2 = y2 – 6y + 9 12. (z + 5)2 = z2 + 10z + 25 13. (2a + 3b) (a – b) = 2a2 + ab – 3b2 14. (a + 4) (a + 2) = a2 + 6a + 8 3x2 2 =1 15. (a – 4) (a – 2) = a – 6a + 8 16. 3x2 3x 3x 3x 2 + 1 3 x 2 1 1 = 2 + 2 =1+ 2 17. 18. = 2 3x + 2 3x + 2 3x 3x 3x 3x 3 3 4 x + 5 4x 5 5 = + =1+ 19. = 20. 4x + 3 4x + 3 4x 4x 4x 4x 21.
7x + 5 7x 5 7x = + = +1 5 5 5 5
EXERCISE 15.1 1. (a) 36.5° C (b) 12 noon (c) 1 p.m, 2 p.m. (d) 36.5° C; The point between 1 p.m. and 2 p.m. on the x-axis is equidistant from the two points showing 1 p.m. and 2 p.m., so it will represent 1.30 p.m. Similarly, the point on the y-axis, between 36° C and 37° C will represent 36.5° C. (e) 9 a.m. to 10 a.m., 10 a.m. to 11 a.m., 2 p.m. to 3 p.m. 2. (a) (i) Rs 4 crore (ii) Rs 8 crore (b) (i) Rs 7 crore (ii) Rs 8.5 crore (approx.) (c) Rs 4 crore (d) 2005 3. (a) (i) 7 cm (ii) 9 cm (b) (i) 7 cm (ii) 10 cm (c) 2 cm (d) 3 cm (e) Second week (f) First week (g) At the end of the 2nd week 4. (a) Tue, Fri, Sun (b) 35° C (c) 15° C (d) Thurs 1 6. (a) 4 units = 1 hour (b) 3 hours (c) 22 km 2 (d) Yes; This is indicated by the horizontal part of the graph (10 a.m. - 10.30 a.m.) (e) Between 8 a.m. and 9 a.m. 7. (iii) is not possible
EXERCISE 15.2 1. Points in (a) and (b) lie on a line; Points in (c) do not lie on a line 2. The line will cut x-axis at (5, 0) and y-axis at (0, 5)
ANSWERS
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3. O(0, 0), A(2, 0), B(2, 3), C(0, 3), P(4, 3), Q(6, 1), R(6, 5), S(4, 7), K(10, 5), L(7, 7), M(10, 8) 4. (i) True (ii) False (iii) True
EXERCISE 15.3 1. (b) (i) 20 km (ii) 7.30 a.m. 2. (a) Yes (b) No
(c) (i) Yes
(ii) Rs 200
(iii) Rs 3500
EXERCISE 16.1 1. 4. 7. 10.
A = 7, B = 6 A = 2, B = 5 A = 7, B = 4 A = 8, B = 1
2. A = 5, B = 4, C = 1 5. A = 5, B = 0, C = 1 8. A = 7, B = 9
3. A = 6 6. A = 5, B = 0, C = 2 9. A = 4, B = 7
EXERCISE 16.2 1. y = 1 4. 0, 3, 6 or 9
2. z = 0 or 9
3. z = 0, 3, 6 or 9
JUST FOR FUN 1. More about Pythagorean triplets We have seen one way of writing pythagorean triplets as 2m, m2 – 1, m2 + 1. A pythagorean triplet a, b, c means a2 + b2 = c2. If we use two natural numbers m and n(m > n), and take a = m2 – n2, b = 2mn, c = m2 + n2, then we can see that c2 = a2 + b2. Thus for different values of m and n with m > n we can generate natural numbers a, b, c such that they form Pythagorean triplets. For example: Take, m = 2, n = 1. Then, a = m2 – n2 = 3, b = 2mn = 4, c = m2 + n2 = 5, is a Pythagorean triplet. (Check it!) For, m = 3, n = 2, we get, a = 5, b = 12, c = 13 which is again a Pythagorean triplet. Take some more values for m and n and generate more such triplets. 2. When water freezes its volume increases by 4%. What volume of water is required to make 221 cm3 of ice? 3. If price of tea increased by 20%, by what per cent must the consumption be reduced to keep the expense the same?
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4. Ceremony Awards began in 1958. There were 28 categories to win an award. In 1993, there were 81 categories. (i) The awards given in 1958 is what per cent of the awards given in 1993? (ii) The awards given in 1993 is what per cent of the awards given in 1958? 5. Out of a swarm of bees, one fifth settled on a blossom of Kadamba, one third on a flower of Silindhiri, and three times the difference between these two numbers flew to the bloom of Kutaja. Only ten bees were then left from the swarm. What was the number of bees in the swarm? (Note, Kadamba, Silindhiri and Kutaja are flowering trees. The problem is from the ancient Indian text on algebra.) 6. In computing the area of a square, Shekhar used the formula for area of a square, while his friend Maroof used the formula for the perimeter of a square. Interestingly their answers were numerically same. Tell me the number of units of the side of the square they worked on. 7. The area of a square is numerically less than six times its side. List some squares in which this happens. 8. Is it possible to have a right circular cylinder to have volume numerically equal to its curved surface area? If yes state when. 9. Leela invited some friends for tea on her birthday. Her mother placed some plates and some puris on a table to be served. If Leela places 4 puris in each plate 1 plate would be left empty. But if she places 3 puris in each plate 1 puri would be left. Find the number of plates and number of puris on the table. 10. Is there a number which is equal to its cube but not equal to its square? If yes find it. 11. Arrange the numbers from 1 to 20 in a row such that the sum of any two adjacent numbers is a perfect square.
Answers 1 2. 212 cm3 2 2 3. 16 % 3
4. 5. 6. 7. 8. 9. 10. 11.
(i) 34.5% (ii) 289% 150 4 units Sides = 1, 2, 3, 4, 5 units Yes, when radius = 2 units Number of puris = 16, number of plates = 5 –1 One of the ways is, 1, 3, 6, 19, 17, 8 (1 + 3 = 4, 3 + 6 = 9 etc.). Try some other ways.
